With the nation’s first presidential primary election less than a week away, pastors across New Hampshire are hoping to surprise the Republican establishment by turning out many thousands of religiously minded voters who have mostly ignored the political world.

The effort had been viewed as a long shot, until Texas Sen. Ted Cruz helped draw thousands of dormant evangelicals to the Iowa caucuses on Monday. Of those who said they were first-time caucusgoers, 61% also described themselves as evangelical Christians, according to entrance polls. Mr. Cruz took the largest proportion of those voters, 34%, on the way to winning the Iowa caucuses.

“I have high hopes here,” said Bryan McCormack, executive director of Cornerstone Policy Research and Action, a conservative New Hampshire nonprofit organization that is working with hundreds of churches across the state to newly mobilize this constituency. “I’ve never seen this type of activity from the faith community regarding voting,” he added.

The Cruz campaign has amassed a network of 200 pastors in churches across the state, said former Sen. Bob Smith, co-head of Mr. Cruz’s campaign in the state. “It’s more elaborate than you think. It’s been overlooked here,” said Mr. Smith, a conservative Republican. “I can assure you it’s big here, not as big as Iowa, but we are working with these people.”